Joint Discussion Paper
Series in Economics
by the Universities of
Aachen ∙ Gießen ∙ Göttingen
Kassel ∙ Marburg ∙ Siegen
ISSN 1867-3678
No. 11-2012
Christoph Bühren, Björn Frank and Rosemarie Nagel
A Historical Note on the Beauty Contest
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A Historical Note on the Beauty Contest
by Christoph Bühren , Björn Frank ♦ (University of Kassel)
and Rosemarie Nagel ∗ (ICREA, BGSE, Universitat Pompeu Fabra)
Updated version: February 20th, 2012
Abstract
Alain Ledoux, who was one of over 6,000 chess players taking part in Bühren and Frank´s
(2012) online Beauty Contest experiment, turned out to be the forgotten inventor of that
game. We reconstruct the birth of the Beauty Contest. In section 1 of our note, its first two
authors outline the history of the game that metamorphosed into the famous guessing game
experiment which was first run in the lab by Rosemarie Nagel. In section 2, Rosemarie Nagel
adds further remarks and thoughts about the development of the experimental Beauty Contest.

c.buehren@uni-kassel.de, University of Kassel, Institute of Economics, Nora-Platiel-Straße 4, 34127 Kassel,
Germany, Tel.: +49-561-804-7267, Fax: +49-561-804-3083
♦
∗
frank@uni-kassel.de
rosemarie.nagel@upf.edu
1. The History of the Beauty Contest Experiment
Rosemarie Nagel (1993, 1995) rightfully became known for the Beauty Contest experiment
that she initially called guessing game. Ho, Camerer and Weigelt (1996, 1998) were the first
to call this guessing game “p-Beauty Contest”, inspired by Nagel´s reference in 1995 to
Keynes' (1936, p. 156) famous comparison of stock market investments and newspaper
beauty contests. The Beauty Contest is an import tool as it provides researchers with a clear
and feasible concept of "depth of reasoning". It is Nagel's achievement to have discovered this
potential of the game, she does not claim to be its inventor - in the initial footnote (Nagel,
1995, p. 1313) she writes: "I learned about the guessing game in a game-theory class given by
Roger Guesnerie, who used the game as a demonstration experiment." Probably Guesnerie's
source was Hervé Moulin (1986) who is the first who published this game in a social science
context. In his textbook, the game was called "Guess the average", with each player picking
an integer between 1 and 999. The game served as the introductory example to the chapter on
successive elimination of dominated strategies. Moulin on his part was inspired by a source
that had slipped his memory when later, in the nineties, consulted by Rosemarie Nagel. Our
recent call for participation in an online Beauty Contest experiment (Bühren and Frank 2012)
finally unearthed Moulin's source.
As a reminder: Participants in Nagel's first experiments (as in Guesnerie’s class) were asked
to guess a (real) number between 0 and 100, the winning number being the one that comes
closest to a share p of the average. In Nagel (1995), p takes the values 1/2, 2/3 and 4/3, with
p = 2/3 being most popular in later replications and variants of the experiment.
In 1981, the French magazine "Jeux & Stratégie", a popular magazine devoted mainly to
strategic board games, but also covering card games and mathematical games, arranged a big
readers' competition consisting of mathematical puzzles but also problems from games such
as chess, bridge and go. Ledoux (1981) reports on almost 15,000 participants, 4,078 of them
being ex aequo, hence the winner had to be decided in a playoff. All first round winners
received a letter with new puzzles, and to avoid another round with multiple winners, chief
editor Alain Ledoux invented in the last question of this letter what is today known as the
Beauty Contest (the name given to it by Ledoux, according to an email to us from July 9th,
2009, was “psycho-statistique”, although this does not appear to have appeared in print).
Readers were asked to state an integer between 1 and 1,000,000,000, the wining number
being the one closest to two third of the average! The average turned out to be
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134,822,738.26, two third of this being 89,881,825.51. This is 8.99 percent of the maximum
number, markedly less than what is typically found in first rounds of Beauty Contest
experiments (Bosch-Domènech, Montalvo, Nagel and Satorra, 2002). However, as explained
above, the participants had been pre-selected, having solved a series of puzzles in the first
round of the contest, and they knew that everyone else was pre-selected. Both facts should
have resulted in the pretty high depth of reasoning. The large interval (with 1,000,000,000
instead of 100 as the upper bound) could also have played a role - an untested hypothesis
suggested to us by Rosemarie Nagel.
In 1983, Jeux & Stratégie held another readers' contest. There was a reference to the first
experiment but not to its result. However, some readers might have remembered or looked up
the result of the previous round. Again, the Beauty Contest was used as a tie break between
equal players on the previous questions - 2898 participants if or reading of Ledoux (1983) is
correct. The target number in 1983 (two third of the average guess) was 67,329,453, or 6.73
percent of the maximum number.
Our preliminary conclusion is that Alain Ledoux should be given some credit for starting this
fascinating line of research, though no one is to blame for the fact that researchers have
previously overlooked him. He did not even sign the articles in Jeux & Stratégie with his
name, and not a single public or scientific library in Germany holds this journal. We would be
glad to receive hints on similar cases already known; the best one that occurred to us so far is
the balanced budget multiplier, typically credited to Haavelmo, but see Gelting (1941),
written in a language alien to almost any economist. 1
1
See also Al Roth in his blog on “Nagel's guessing/beauty contest game: a famous experiment in game
theory” where he quotes from Roth and Sotomayor (1990, p.170): "What is important about Columbus'
discovery of America is not that it was the first, but that it was the last. After Columbus, America was never lost
again."
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2. Some Further Remarks by Rosemarie Nagel 2
1. In November 1990 in the LSE Tore Ellingsen presented a one shot guessing game in
his master student IO class. I chose 22, according to 50*2/3*2/3. I did not further think
about this game.
2. Shortly thereafter I saw the game again in Gueneries’ Phd game theory class, like
some other students who were also in Tore’s class. I chose a number a bit lower than
in the other class and won. Then Guenerie asked us to play again. And again I won. I
also saw the choices and saw some numbers near and at 33 and 22. I asked Guenerie
for the data set and analyzed it, and I also did a pilot in LSE. …. I find this important
to add because it shows that sometimes you need to have been your own subject to see
the beauty of some idea. Remember, the game was around for some years, but nobody
thought it was interesting to do an experiment. First of all at that time grade students
were not interested/educated in doing experimental economics. And the theorists only
saw that behaviour is not in equilibrium, which is of course trivial. I needed two inputs
of classes to see its beauty. At the time I was already a researcher-phd student in
experimental economics and searching for a topic far away from ultimatum games
which I had done in my first paper. Guenerie used this game as a demonstration
experiment to show that rationalizability doesn’t work in this game while I used this
game to test the 50*pn model, which I had actually used in Tore Ellingsen's class.
However, I needed this second class by Guenerie to appreciate the game.
3. One Phd student, Klaus Kultti, like me in LSE in 1991, now Helsinki University,
pointed out that the game (maybe also my reasoning process) is due to Keynes beauty
contest.
4. Reinhard Selten, my supervisor, was against the name beauty contest as Keynes´
contest has multiple equilibria and so we named it guessing game. (John Duffy
insisted to call it Keynes beauty contest in Duffy and Nagel (EJ 1997)), which indeed
is also correct since it is just a special case with the parameter equal 1 (1*average).
5. In Pittsburgh in 1995 Oliver Schulte a graduate student in Computer science and
philosophy in Carnegie Mellon (now computer science prof.) found the guess the
average game in Moulin’s book which Oliver used for teaching purposes.
2
See also Coricelli and Nagel (2010) for a personal account on the guessing game history.
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6. I saw in 1995 in a public choice conference (I think) Herve Moulin and told him that I
had done experiments in his game. He then referred me to Pour la Science (French
version of Scientific American) as his original source, which now actually turned out
to be Jeux & Stratégie.
Now finally the game has found his founding father, who, however, saw no special
interest in it besides providing a tie breaker.
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References
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